Distribution of energy within the spectrum. The Stefan-Boltzmann law gives the total radiation emitted by the body but tells us nothing about how it is distributed across the spectrum. If measurements are made of the energy emitted at different temperature for a black body then a series of curves like those shown in Figure 1 can be obtained. Here the vertical axis shows the energy density (that is the energy emitted per square metre per second in a small wavelength range from λ to λ+dλ) and the horizontal axis shows the wavelength. E
T = 4000 K
T = 3700 K
500 1000 Wavelengthnm Figure 1
Some important facts can be deduced from these curves: (a) the area between any energy-wavelength curve and the wavelength axis gives the total energy emitted by the body per unit area at that temperature (b) the maxima of the curves moves towards short wavelengths at higher temperatures (c) the curves for lower temperatures lie completely inside those of higher temperature
It was found that:
(a) λmT = constant where λm is the wavelength at which most energy is emitted, that is at the maxima of the curves. The constant has a value of 2.898x10-3m K. 5 (b) the energy emitted at this wavelength (λm) is proportional to T .
o Where T is the temperature of the body in Kelvin (TK= C+273.15)
These two results are known as Wien's laws
A range of values for λm is shown in the table Temperature /K Wavelength (λm) /nm below. It must be remembered that these are for 500 5800 true black bodies only. 750 3900 For an electric light bulb the filament 1000 3000 temperature (T) = 1750 K and the peak of the 1750 1650 6000 480 energy-wavelength curve (λm) is at 1400 nm. 30000 97
1 You can see that if λm for a black body lies in the red region of the spectrum the body will appear red hot, and as it gets hotter this peak will move towards the violet end of the spectrum. However this does not mean that the body will look “violet hot”. The reason for this is that at the higher temperature all visible wavelengths will be present to some extent and so the body will appear “white hot”. A white-hot body will give high emission across the whole range of the visible spectrum.
If we know the value of λm for one black body at a known temperature we can use Wien's law to calculate the temperature of another black body providing the wavelength at which maximum energy is emitted is known. This has been used extensively in astronomy for finding the temperatures of stars.
Example problem A black body (the Sun) with a surface temperature of 6000 K emits radiation with λ m = 420 nm. Calculate the temperature of Sirius (the brightest star in the northern skies) if λm for Sirius is 72 nm. -9 -9 λm1T1 = λm2T2 therefore 420x10 x6000 = 72x10 xT2 Therefore: T2 = 35 000 K
2